Q1(l). Meaning and dimensions of |Ψ(x)|²

Answer in brief

|Ψ(x)|² is the probability density: |Ψ(x)|² dx is the probability of finding the particle between x and x + dx; in one dimension it has dimensions of length⁻¹ (m⁻¹).

Key points

• Born interpretation: Ψ itself may be negative or complex; |Ψ|² = Ψ*Ψ is real and ≥ 0: the probability density.
• Probability in a finite range: P(x₁ → x₂) = ∫ |Ψ|² dx from x₁ to x₂ (area under the curve).
• Normalisation: ∫ |Ψ|² dx over all space = 1 ⇒ Ψ must be square-integrable.
• The integral is a pure number and dx is a length ⇒ [|Ψ(x)|²] = L⁻¹ (m⁻¹); Ψ has m⁻¹ᐟ²; in three dimensions |Ψ|² has m⁻³.